||matthewissac (at) rutgers (dot) edu
||Rutgers University - New Brunswick
||Algebraic Invariants of Pretzel Knots
About My Project
Heegaard Floer Homology allows us to associate a knot to a doubly filtered chain complex. Maps between such complexes are called chain maps. We are interested in an involutive chain map, iota_k, and its computation for pretzel knots P(-2,m,n) with m and n odd. This map has been computed for a torus knots and we hope our computation will help provide more examples.
I was given a brief introduction to knot theory, knot invariants, and some basic homological algebra. Some basic knot invariants I learned about are: three colorability, Alexander polynomial, and Heegaard Floer Homology. I was given a crash course in computing homologies and had plenty of examples to practice with. Moreover, I learned how the Euler Characteristic generalized to bi-graded homologies and how this information encompasses the Alexander polynomial. Finally, I capped the week off by looking into an introductory paper to Heegaard Floer Homology.
We were introduced to filtered chain complexes and chain maps. Later on, we saw the Sarkar Involution an involutive, filtered, chain map. This is a crucial piece of data for iota_k, since iota_k^2 is chain homotopic to the Sarkar involution. We also worked through an example of a computation of iota_k for the figure 8 knot.
We saw the technical side of CFK^∞, the chain complex we have been working with. We also saw how the generators of this complex come from intersection points on Heegaard diagrams.
We have started to read Knot Floer Homology of (1,1)-knots by Goda, Matsuda, and Morifuji. They have already computed CFK^∞ for P(-2,m,n) with m and n odd. We started to understand their notation and worked out what CFK^∞ is for P(-2,5,5) and worked out iota_k for this knot.
I began working on a calculator which would accept integers n and m and compute the shape of the complex. There were initally many bugs but I eventually ironed it out and we can now compute more examples.
I have begun thinking about a change of basis to do to the complex and how to create a box and staircase pattern.
I am beginning to understand how the indices relate to one another, in particular I found certain conditions on when two generators lie in the same filtration level.
We have begun discussing the main theorem, in addition to new invariants. We have drafted this and are now preparing our final presentation.
Finished our presentation but still more work to be done. We still need to prove main theorem and finish the final change of basis.
References & Links
After the program finished, we proved our main theorem. All that remains is the paper, which is currently being written.
Here are the papers I have read for my project:
Knot Floer homology of (1,1)-knots.
Goda, Matsuda, and Morifuji - ..
An introduction to Heegaard Floer homology
Ozsvath and Szabo - Retrieved from here..
Here is my mentor's websites, and the REU website:
- My Mentor's Website
- The REU Website
HTML 4 font rendering:
∂y/∂t = ∂y/∂x √2 =1.414
If f(t)= ∫t 1 dx/x then
f(t) → ∞ as t → 0. This really means:
(∀ε ∈ℝ, ε>0) (∃δ>0)
f(δ) > 1/ε .
ℕ (natural numbers), ℤ (integers), ℚ (rationals),
ℝ (reals), ℂ (complexes)